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Bartlett's formula for a general class of nonlinear processes

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  • Christian Francq
  • Jean‐Michel Zakoïan

Abstract

. A Bartlett‐type formula is proposed for the asymptotic distribution of the sample autocorrelations of nonlinear processes. The asymptotic covariances between sample autocorrelations are expressed as the sum of two terms. The first term corresponds to the standard Bartlett's formula for linear processes, involving only the autocorrelation function of the observed process. The second term, which is specific to nonlinear processes, involves the autocorrelation function of the observed process, the kurtosis of the linear innovation process and the autocorrelation function of its square. This formula is obtained under a symmetry assumption on the linear innovation process. It is illustrated on ARMA–GARCH models and compared to the standard formula. An empirical application on financial time series is proposed.

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  • Christian Francq & Jean‐Michel Zakoïan, 2009. "Bartlett's formula for a general class of nonlinear processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 30(4), pages 449-465, July.
    • Handle: RePEc:bla:jtsera:v:30:y:2009:i:4:p:449-465
    DOI: 10.1111/j.1467-9892.2009.00623.x
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    1. Ling, Shiqing & McAleer, Michael, 2002. "NECESSARY AND SUFFICIENT MOMENT CONDITIONS FOR THE GARCH(r,s) AND ASYMMETRIC POWER GARCH(r,s) MODELS," Econometric Theory, Cambridge University Press, vol. 18(3), pages 722-729, June.
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    5. Kokoszka, Piotr S. & Politis, D N, 2008. "The Variance of Sample Autocorrelations: Does Barlett's Formula Work With ARCH Data?," University of California at San Diego, Economics Working Paper Series qt68c247dp, Department of Economics, UC San Diego.
    6. Christian Francq & Jean-Michel Zakoïan, 2008. "Barlett’s Formula for Non Linear Processes," Working Papers 2008-05, Center for Research in Economics and Statistics.
    7. Lütkepohl,Helmut & Krätzig,Markus (ed.), 2004. "Applied Time Series Econometrics," Cambridge Books, Cambridge University Press, number 9780521839198.
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    7. BAUWENS, Luc & HAFNER, Christian & LAURENT, Sébastien, 2011. "Volatility models," LIDAM Discussion Papers CORE 2011058, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
      • Bauwens, L. & Hafner, C. & Laurent, S., 2012. "Volatility Models," LIDAM Reprints ISBA 2012028, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
      • Bauwens, L. & Hafner C. & Laurent, S., 2011. "Volatility Models," LIDAM Discussion Papers ISBA 2011044, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
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    More about this item

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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